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Canceling branch points on projections of surfaces in $ 4$-space

Authors: J. Scott Carter and Masahico Saito
Journal: Proc. Amer. Math. Soc. 116 (1992), 229-237
MSC: Primary 57Q35; Secondary 57Q45
MathSciNet review: 1126191
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Abstract: A surface embedded in $ 4$-space projects to a generic map in $ 3$-space that may have branch points--each contributing $ \pm 1$ to the normal Euler class of the surface. The sign depends on crossing information near the branch point. A pair of oppositely signed branch points are geometrically canceled by an isotopy of the surface in $ 4$-space. In particular, any orientable manifold is isotopic to one that projects without branch points. This last result was originally obtained by Giller. Our methods apply to give a proof of Whitney's theorem.

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  • [1] T. F. Banchoff, Double tangency theorems for pairs of submanifolds, Geometry Symposium Utrecht 1980 (Looijenga, Seirsma, and Takens, eds.), Lecture Notes in Math., vol. 894, Springer-Verlag, Berlin and New York, 1981, pp. 26-48. MR 655418 (83h:53005)
  • [2] -, Normal curvature and Euler classes for polyhedral surfaces in $ 4$-space, Proc. Amer. Math. Soc. 92 (1984), 593-596. MR 760950 (86c:57019)
  • [3] -, The normal Euler class of a polyhedral surface in $ 4$-space, unpublished notes and letters.
  • [4] T. F. Banchoff and Ockle Johnson (to appear).
  • [5] S. E. Cappell and J. L. Shaneson, An introduction to embeddings, immersions and singularities in codimension two, Proc. Sympos. Pure Math., vol. 32, Amer. Math. Soc., Providence, RI, 1978. MR 520529 (80e:57013)
  • [6] Cole Giller, Towards a classical knot theory for surfaces in $ {{\mathbf{R}}^4}$, Illinois J. Math. 26 (1982), 591-631. MR 674227 (84c:57011)
  • [7] Seiichi Kamada, Non-orientable surfaces in $ 4$-space, Osaka J. Math. 26 (1989) 367-385. MR 1017592 (91g:57022)
  • [8] Dennis Roseman, Projections of codimension two embeddings, preprint. MR 1865719 (2002j:57042)
  • [9] -Reidemeister-type moves for surfaces in four dimensional space, preprint.
  • [10] Bruce Trace, A general postion theorem for surfaces in $ 4$-space, Comtemp. Math., vol. 44, Amer. Math. Soc., Providence, RI, 1985, pp. 123-137. MR 813108 (87f:57005)
  • [11] H. Whitney, On the topology of differentiate manifolds, Lectures in Topology (Raymond L. Wilder and William L. Ayres, eds.), Univ. of Michigan Press, Ann Arbor, 1941. MR 0005300 (3:133a)

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Keywords: Embedded surfaces in $ 4$-space, projections, branch points, normal Euler number
Article copyright: © Copyright 1992 American Mathematical Society

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